Ideal solution phase diagram computation

[megid]

Homework Statement

Hi there, I'm very new here and I was hoping someone would be able to help me out a bit! I have a few questions but right now I'm going to start with this problem.

Elements A and B form ideal solutions in both liquid and solid phases. Compute the A-B phase diagram using the following data:

Element A: Melting point = 1000 (K), Heat of fusion (J/mol) = 8314
Element B: Melting point = 800 (K), Heat of fusion (J/mol) = 6651

Homework Equations

Some equations that I tried to figure out to use would be
H=TS
to calculate the entropy of tranformation from the heat of fusion at the melting temperature
K1 = e^-(S1*(T(1000)-T)/RT)
K2 = e&-(S2*(T(800)-T)/RT)
X2alpha = (K1-1)/(K1-K2)
X2beta = K2*X2alpha

The Attempt at a Solution

What I tried to do was calculate the entropy of formation first, then calculate G from the melting temperature range 800-1000. I then tried to calculate the K values, and accordingly, the X values as well... but my numbers seemed a little crazy. I'm pretty sure I have to plot the G value vs composition at an interval of temperatures to start plotting my phase diagram, but I'm at a loss right now. Could someone please help!

 

[anathema]

Hi there, I would be happy to help you with this problem. To start, let's review some important concepts and equations that will help us solve this problem.

1. Ideal Solutions: In an ideal solution, the enthalpy of mixing (ΔH_mix) is equal to zero. This means that there is no energy required to mix the two elements together. Therefore, the Gibbs free energy of mixing (ΔG_mix) is also equal to zero.

2. Entropy (S): Entropy is a measure of disorder or randomness in a system. It is given by the equation S = Q/T, where Q is the heat transferred and T is the temperature.

3. Gibbs Free Energy (ΔG): The Gibbs free energy is a measure of the amount of energy that is available to do work in a system. It is given by the equation ΔG = ΔH - TΔS, where ΔH is the enthalpy and ΔS is the entropy.

Now, let's apply these concepts to solve the problem. We know that the elements A and B form ideal solutions in both liquid and solid phases. This means that the enthalpy of mixing (ΔH_mix) is equal to zero. Therefore, the Gibbs free energy of mixing (ΔG_mix) is also equal to zero.

We also know the melting points and heat of fusion for both elements A and B. We can use this information to calculate the entropy of transformation (ΔS_trans) from the heat of fusion at the melting temperature using the equation ΔS_trans = ΔH_fusion/T_melt. This will give us the entropy change when the elements melt.

Next, we need to calculate the Gibbs free energy of fusion (ΔG_fusion) for both elements A and B. We can use the equation ΔG_fusion = ΔH_fusion - TΔS_trans. This will give us the energy required to melt the elements at their respective melting points.

Now, we can use the Gibbs free energy values to calculate the equilibrium constant (K) using the equation K = e^(-ΔG/RT). This will give us the ratio of the concentrations of the liquid and solid phases at equilibrium.

Finally, we can use the equilibrium constant values to calculate the composition of the liquid and solid phases at different temperatures using the equations X_liquid = K/(1+K) and X_solid = 1/(